J. theor. Biol. (2001) 210, 000}000 doi:10.1006/jtbi.2001.2304, available online at http://www.idealibrary.com on
E4ects of Habitat Destruction and Resource Supplementation in a Predator}Prey Metapopulation Model
ROBERT K. SWIHART*-,ZHILAN FENG?,NORMAN A. SLADEA,DORAN M. MASON* AND THOMAS M. GEHRING*
*Department of Forestry and Natural Resources, Purdue ;niversity, =est ¸afayette, IN 47907-1159, ;.S.A., -Department of Mathematics, Purdue ;niversity, =est ¸afayette, IN 47907-1395, ;.S.A. and ?Department of Ecology and Evolutionary Biology and Natural History Museum and Biodiversity Research Center, ¹he ;niversity of Kansas, ¸awrence, KS 66045-2454, ;.S.A.
(Received on January 2000, Accepted in revised form on 2000)
We developed a mean "eld, metapopulation model to study the consequences of habitat destruction on a predator}prey interaction. The model complements and extends earlier work published by Bascompte and SoleH (1998, J. theor. Biol. 195, 383}393) in that it also permits use of alternative prey (i.e., resource supplementation) by predators. The current model is stable whenever coexistence occurs, whereas the earlier model is not stable over the entire domain of coexistence. More importantly, the current model permits an assessment of the e!ect of a generalist predator on the trophic interaction. Habitat destruction negatively a!ects the equilibrium fraction of patches occupied by predators, but the e!ect is most pronounced for specialists. The e!ect of habitat destruction on prey coexisting with predators is dependent on the ratio of extinction risk due to predation and prey colonization rate. When this ratio is less than unity, equilibrial prey occupancy of patches declines as habitat destruction increases. When the ratio exceeds one, equilibrial prey occupancy increases even as habitat destruction increases; i.e., prey &&escape'' from predation is facilitated by habitat loss. Resource supple- mentation reduces the threshold colonization rate of predators necessary for their regional persistence, and the bene"t derived from resource supplementation increases in a nonlinear fashion as habitat destruction increases. We also compared the analytical results to those from a stochastic, spatially explicit simulation model. The simulation model was a discrete time analog of our analytical model, with one exception. Colonization was restricted locally in the simulation, whereas colonization was a global process in the analytical model. After correcting for di!erences between nominal and e!ective colonization rates, most of the main conclusions of the two types of models were similar. Some important di!erences did emerge, however, and we discuss these in relation to the need to develop fully spatially explicit analytical models. Finally, we comment on the implications of our results for community structure and for the conservation of prey species interacting with generalist predators. ( 2001 Academic Press
1. Introduction 1997; With & King, 1999) are widespread in natu- ral systems due to anthropogenic changes in land Habitat loss and habitat fragmentation (sensu use (e.g., Saunders et al., 1991; Andersen et al., MoK nkkoK nen and Reunanen, 1999; With et al., 1996). Changes in the composition and physiog- nomy of a landscape resulting from habitat loss -Author to whom correspondence should be addressed. and fragmentation (Dunning et al., 1992) can
0022}5193/01/000000#00 $35.00/0 ( 2001 Academic Press
JTBI 20012304 PROD.TYPE: TYP ED: Mangala Gowri PAGN: RS: TJ SCAN: Anitha pp. 1}17 (col.fig.: NIL) 2 R. K. SWIHART E¹ A¸. alter genetic structure (Gaines et al., 1997), indi- interspeci"c interactions which change the vidual behavior (Lima & Zollner, 1996; Sheperd abundance, distribution, and persistance of a spe- & Swihart, 1995), local population dynamics cies (Billick & Case, 1994). Theoretical studies (Nupp & Swihart, 1996, 1998), interspeci"c inter- have demonstrated the potential for fragmenta- actions (Keyser et al., 1998), and community tion to produce higher-order e!ects among com- composition (Dunstan & Fox, 1996; Hecnar & petitors (Tilman et al., 1994; Moilanen & Hanski, M'Closkey, 1997; Kolozsvary & Swihart, 2000). 1995; Nee et al., 1997; Huxel & Hastings, 1998) Not surprisingly, habitat destruction has been and predators and prey (May, 1994; Kareiva implicated as the major threat to biological & Wennergren, 1995; Holyoak & Lawler, 1996; diversity (Wilcox & Murphy, 1985). Nee et al., 1997; de Roos et al., 1998). The metapopulation concept provides a useful Habitat loss and fragmentation are of particu- framework within which to study the implica- lar concern to conservation biologists in the con- tions of habitat loss and fragmentation. A meta- text of extinction thresholds (With & King, 1999); population is viewed as a network of idealized i.e., nonlinear responses of populations to habitat patches (fragments) in which species habitat loss which lead to abrupt declines in occur as discrete local populations connected by patch occupancy over a narrow range of habitat dispersal (Hanski, 1998). In its original formula- destruction. Few metapopulation models of tion (Levins, 1969), a proportion p of all patches predator}prey systems have incorporated a com- are occupied, with empty patches being colonized ponent of habitat loss (but see Kareiva & at rate c and occupied patches going extinct at Wennergren, 1995). rate e: Recently, Bascompte & SoleH (1998) formulated a Levins-type metapopulation model to examine the e!ect of habitat destruction on the dynamics dp of a prey and its specialist predator. They showed "cp(1!p)!ep. dt that predators were more sensitive to habitat fragmentation than were prey, and that extinc- tion thresholds for predators were related to Under these conditions, the equilibrium propor- predator colonization rate. Although the results tion of occupied patches, p*, in a metapopulation of Bascompte & SoleH (1998) are interesting, they is determined by the per patch probabilities of did not include an analysis of the stability of the colonization and extinction; i.e., p*"1!(e/c). model's equilibria. In addition, many predators Of course c and e are in#uenced by factors intrin- in natural systems are not obligate specialists but sic to the organism under study (e.g., vagility, rather are capable of relying upon other re- territoriality, population density, variation in de- sources to meet their energetic needs. Thus, many mographic rates) and by factors related to the predators are capable of resource supplementa- landscape or patch (e.g., patch isolation, patch tion (Dunning et al., 1992) to varying degrees, area, patch orientation, patch geometry). Gener- and this may have important implications for the alizations are possible regarding the e!ects of dynamics of a predator}prey system in a frag- some of these factors (Fahrig & Paloheimo, 1988; mented landscape. Herein, we revisit the model of AndreH n, 1994; Frank & Wissel, 1998; Wol!, Bascompte & SoleH (1998), pose an alternative 1999), and considerable progress has been made formulation, and relax the assumption of extreme in modifying the Levins single-species model for specialization by the predator. Speci"cally, predictive purposes in real landscapes (reviewed our objectives are to: (1) examine the stability in Hanski, 1998). conditions for the model developed by Bas- Less attention has been paid to metapopula- compte & SoleH (1998); (2) formulate an alterna- tion models of interacting species, despite the tive model based on random encounter probabil- strong likelihood that asymmetric e!ects of ities of predator and prey; and (3) examine the habitat fragmentation could alter dramatically dynamics of predator and prey under varying the strength, and perhaps even the type, of inter- conditions of resource supplementation by the actions. Higher-order e!ects refer to modi"ed predator.
JTBI 20012304 PREDATOR}PREY METAPOPULATION 3
2. The Bascompte and SoleH Metapopulation Model able for colonization (see also Kareiva & Wen- nergren, 1995). The resulting model is as follows: Bascompte & SoleH (1998) relied upon a mean "eld model; i.e., a model depicting behavior in a homogeneous mixing metapopulation composed dx "c x(1!x!D)!e x!ky, (1a) of an in"nite number of local populations. Fol- dt V V lowing May (1994), an additional assumption was that predators were specialists and thus dy "c y(x!y)!e y. (1b) could exist only on patches containing the prey in dt W W question. Let x and y represent the proportion of patches occupied by prey and predators, respec- Because the predator's occurrence in a patch is tively. Then the extension of the Levins (1969) conditional on the prey's occurrence there, incor- metapopulation model to two trophic levels by poration of D is only required for the prey eqn Bascompte & SoleH (1998) is given as (1a). Bascompte & SoleH (1998) examined the be- havior of this predator}prey model by noting the dx k "c x(1!x)!e x!ky, e!ect of D, cW, and on the equilibrial fraction of dt V V patches containing prey (x*) and predators (y*). They also examined a spatially explicit form of dy the model using a cellular automaton, thereby "c y(x!y)!e y. dt W W assessing the robustness of the analytical model to the incorporation of local spatial structure. We Note that the equation for the prey di!ers from will revisit portions of their analysis in Section 6. the original Levins (1969) formulation in having an additional term, ky. Bascompte & SoleH (1998) added this term to represent the additional 3. An 99Ignorant Predator:: Metapopulation Model extinction risk imposed on prey in patches also In the model formulated by Bascompte & SoleH occupied by predators. That is, eV represents the (1998), cW represents the per patch rate at which per patch rate of extinction of prey independent predators colonize a &&habitable'' site; i.e., a patch of the e!ect of predators, and k represents the containing prey of type X. However, predators additional rate of prey extinction due to pred- often must deal with imperfect information re- ators on the fraction of patches (y) in which they garding their environment, which frequently can co-occur. Thus, the total extinction rate of prey result in suboptimal movements (e.g., Zollner, #k on patches also occupied by predators is eV . 2000); i.e., movements to patches without prey of Bascompte & SoleH (1998) restricted composite type X. The degree to which predators can track rates for extinction (and colonization) to the in- the distribution of prey is dependent upon nu- terval from 0 to 1, permitting their interpretation merous factors, including the sensory capabilities as probabilities of occurrence in dt; we have re- of the predator, behavioral or ecological charac- tained this convention in our paper. teristics of the prey that alter their detectability, The assumption that predators are specialists and characteristics of the physical environment capable of surviving only on patches with prey of (Mason & Patrick, 1993; Brown et al., 1999). In type X also alters the equation for the predator eqn (1b), predators colonize sites containing prey relative to the original formulation of Levins of type X at a rate cW. An alternative scenario is to (1969). Speci"cally, if some fraction y of patches recognize that predators make mistakes when is occupied by the predator (and, by extension, acting without perfect information regarding the prey type X), then only a fraction x!y of distribution of prey. As an extreme example, sup- patches remains available for colonization by the pose that predators know nothing regarding the predator. distribution of X-type prey in the landscape. In To model habitat destruction, Bascompte other words, predator colonization of a patch occ- & SoleH (1998) introduced a term, D, representing urs independently of whether it is occupied by an the fraction of sites destroyed and thus unavail- X-type prey. This produces a random-encounter
JTBI 20012304 4 R. K. SWIHART E¹ A¸. model, which we refer to as the &&ignorant pred- tively, when t"0, predators are functionally ator'' model to highlight the fact that coloniz- independent of X-type prey, consistent with a sys- ation of a site by a predator is not conditional on tem in which predation on X-type prey occurs the occurrence of X-type prey. In the Appendix incidental to primary foraging pursuits of gener- we show the equivalence of the ignorant predator alist predators (i.e., incidental predation, sensu model to a model formulated in terms of state Vickery et al., 1992; Schmidt & Whelan, 1998). transitions of patches. The ignorant predator In Section 6 we derive the equilibria for this model is as follows: ignorant-predator model, analyse the general conditions for stability, and examine the behav- dx ior of the model in response to changes in habitat "c x(1!x!D) dt V destruction (D), predator colonization rate (cW), extinction rate of prey due to predation (k), and extinction rate of predator due to ignorance of ! ! ! #k eVx(1 y) (eV )xy, (2a) the location of X-type prey (t). We also assess the robustness of the ignorant predator model dy" ! ! to variation in local structure of the landscape by cWy(1 y D) dt comparing results to those produced by its spa- tially explicit analog. First, though, we compare ! ! #t ! eWxy (eW )(1 x)y. (2b) more closely the formulations of the ignorant predator model and the model of Bascompte The equations for both prey and predator con- & SoleH (1998). We then provide the conditions tain positive colonization terms. The coloniz- necessary for equivalence of the two models. ation term for the prey is identical to the term in eqn (1a). However, the term in eqn (2b) re#ects the probability of predator colonization of any 4. Comparison of Metapopulation Models extent patch without a predator, including Algebraic manipulation results in a simpli"ed ! ! patches without X-type prey (i.e., 1 y D). In form of the ignorant predator model from eqns eqn (2a) we have decomposed the probability of (2a) and (2b) as follows: extinction of prey into two terms. A fraction x(1!y) of patches are occupied only by prey, and prey on these patches have a per-patch ex- dx" ! ! ! !k cVx(1 x D) eVx xy, (3a) tinction probability of eV. The remaining patches dt occupied by prey also are occupied by predators. Thus, a fraction xy of patches exhibit the additive dy" ! ! ! !t ! extinction probabilities intrinsic to prey and due cWy(1 y D) eWy y(1 x). (3b) #k dt to predation (eV ). To facilitate interpretation we also have decomposed the probability of ex- tinction of predators into two terms in eqn (2b). For simplicity, assume D"0. Comparing eqns In the fraction of patches occupied by both (1a) and (3a), the only di!erence in the prey equa- X-type prey and the predator (xy), predator ex- tions for the two models resides in the last term. tinction occurs with probability eW. In patches For the ignorant predator model (3a), the rate of without X-type prey, predators pay an added increase of x is reduced by an amount k in the cost (t) in terms of increased probability of local fraction xy of patches in which both prey and extinction for mistakenly colonizing an inferior predator reside. In the model of Bascompte t" ! resource patch. When 1 eW, the instan- & SoleH (1998), co-occurrence of predator and taneous probability of predator extinction is 1 on prey is not explicitly addressed, because the a patch with no X-type prey, consistent with an occurrence of predators is conditional on prey. extreme specialist, but di!ering from the model of Thus, equivalence of the two prey equations is Bascompte & SoleH (1998) by allowing coloniz- predicated on the equivalence of the last terms; ation of patches lacking X-type prey. Alterna- namely, ky,kxy. Likewise, equivalence of the
JTBI 20012304 PREDATOR}PREY METAPOPULATION 5 ignorant predator [eqn (3b)] with the omniscient ing patch occupied by species i could colonize the predator of the Bascompte & SoleH (1998) model focal patch with probability cG. Adopting this rule "t (1b) requires that cW in eqn (3b). ensured that colonization was a local process. Moreover, independence of colonization prob- abilities among patches resulted in a functional probability of colonization that varied with the 5. A Spatially Explicit Predator}Prey number of neighboring patches occupied by i. Metapopulation Model Speci"cally, the probability of colonization of ! ! L To determine how local colonization processes a patch in the explicit model is 1 (1 cG) , in#uence the dynamics of the ignorant-predator where n is the number of neighboring patches system, we developed a spatially explicit simula- occupied by species i,0)n)4. After the state of tion model. Following Bascompte & SoleH (1998), a patch had been updated to account for coloniz- we constructed a stochastic cellular automation ation events, the state was saved to the new with four nearest neighbors coupling. We used lattice. a 100;100 lattice of patches and incorporated Our interest in simulation was to compare habitat destruction by randomly removing a spe- results of our analytical model (3), in which dis- ci"ed fraction of patches from those considered persal occurs globally, with a model in which to be usable, i.e., 1!D. After categorizing each dispersal was constrained to occur locally. Thus, patch as either available or destroyed, pred- we conducted simulations until a steady state was ator}prey dynamics were modeled as described attained in the fraction of available patches occu- below. The complete set of the state transitions is pied by prey and predator. A steady state was provided in Appendix A. assumed to occur when three iterations yielded Initially, prey and predators were distributed a change of )0.0001 in the running average of randomly and independently among half of the patch occupancy (excluding the "rst 20 iterations available patches. Thus, approximately of avail- to reduce the in#uence of initial conditions). able patches were occupied by both species Because of the discrete nature of the simulations, at the beginning of a simulation. Extinction equilibrial values conceivably could be in#u- and colonization processes were applied stochas- enced by the timing of the census of patches tically on a patch-by-patch basis. The state relative to the life cycles of the populations (e.g. of each available patch (empty, occupied by Caswell, 1989). Thus, we computed equilibrial X-type prey, occupied by predator, occupied by values based on the average of census conducted both species) and its four nearest neighbors before and after colonization. Results presented determined the particular probabilities used below are averages of three replicate runs for (Appendix A). each set of parameter values. If a patch was occupied only by prey, extinc- tion of prey occurred with probability eV. How- ever, if a patch contained both species, extinction 6. Results #k of prey occurred with probability eV and extinction of the predator occurred with prob- 6.1. EQUILIBRIA AND STABILITY ANALYSIS ability eW. If no X-type prey currently occupied As noted by Bascompte & SoleH (1998), there the patch, extinction of the predator occurred are three equilibria for the model in eqn (1). The #t " " with probability eW . After the state of a patch "rst two, E (0, 0) and E (x, 0), are bound- had been updated to account for extinction ary equilibria. Denote the critical fractions of events, the state was saved to a new lattice for use habitat destruction at which predator and prey in determining colonization. become extinct as D and D , respectively. E is A A After extinctions has been determined for the stable if D'D "1!(e /c ) and unstable if A V V entire lattice, colonization also was modeled D(D . For E , x "1!D!(e /c ), and thus A V V stochastically. If the patch was unoccupied by E exists if and only if D(D . E is stable if A species i, a check was made of the state of each of D'D " 1!e /c !e /c and unstable if A V V W W its four-nearest neighboring patches. A neighbor- D(D . The third equilibrium, E*"(x*, y*), is A
JTBI 20012304 6 R. K. SWIHART E¹ A¸. an interior equilibrium with x* and y* given by at E*G is given as (Bascompte & SoleH , 1998) !c x* !kx* J*" V G G . i A ty* !c y*B " 1 C# C# k eW G W G x* C A 4cV B D , (4a) 2cV cW For a 2;2 Jacobian, J, stability exists if the determinant, Det(J)'0 and the trace, Tr(J)(0 e " y*"x*! W . (4b) (Gurney & Nisbet, 1998). For Ji*, Det(Ji*) c #kt ' "! ! W (cVcW )x*G y*G 0 and Tr(J*i ) cVx*G ( cWy*G 0. Hence, E*G is always stable when it C" ! ! !k exists, in contrast to the interior equilibrium from In eqn (4a), cV (1 D) eV . Because of the conditional nature of y on x, E* exists if and the model of Bascompte & SoleH (1998). only if D(D . However, it can be shown (Ap- A pendix A) that there exist critical values DA and 6.2. EFFECTS OF HABITAT DESTRUCTION AND k ( k'k RESOURCE SUPPLEMENTATION A such that E* is unstable for D DA and A, ' k(k or for D DA and A. The region of instabil- Increasing the level of habitat destruction al- ity can be substantial for reasonable parameter ways leads to a smaller fraction of patches occu- values. pied by predators at equilibrium in the ignorant- Four possible equilibria exist for the ignorant- predator model (Appendix A, Figs 1 and 2). The predator model in eqn (2). The "rst two, E and use of alternative resources by a predator in the E, are identical to those discussed previously in ignorant-predator model is capable of counter- the model of Bascompte & SoleH (1998). A third acting some of the negative e!ects of habitat " boundary equilibrium, E (0, y), exists be- destruction on predator persistence. Speci"cally, cause predators are no longer constrained to resource supplementation by predators increases occur only on patches containing X-type prey. the proportion of additional resource patches Thus, at E, X-type prey are extinct, but pred- that they can exploit, albeit with varying degrees ators persist due to resource supplementation. of e$ciency, from patches occupied only by X- The equilibrium fraction of patches occupied type prey (x!y) to all undestroyed patches by predators in the absence of X-type prey is (1!D!y). For this condition to be true, we " ! ! #t y 1 D (eW )/cW. Existence of E and must assume that all intact patches have equal E can be expressed in terms of critical values amounts of some resource(s) other than X-type of habitat destruction: E exists if and only if prey. Thus, resource supplementation permits D(D "1!(e /c ), and E exists if and only if V V V predators to dilute the e!ect of habitat destruc- D(D "1!(e #t)/c . A complete stability W W W tion by potentially accessing an additional frac- analysis of these boundary equilibria is provided tion 1!D!x of patches. From the perspective in Appendix A. of foraging ecology, the probability of predator The fourth equilibrium of the ignorant-pred- survival in a patch without X-type prey is related " ator model is given as E*G (x*G , y*G ), where to the relative e$ciency with which alternative resources in the patch can be used by predators ! !t 1 and is measured by 1 eW . As predators x*" (c (c !k)(1!D)!c e #k(e #t)), G b W V W V W become less dependent on X-type prey for their survival in a patch (lower t), the equilibrium (5a) proportion of patches occupied increases for a given level of habitat loss (Figs 1 and 2). 1 y*" (c c (1!D)!c tD!c e !e t) (5b) For prey, the interactive e!ects of habitat de- G b V W V V W V struction and resource supplementation by pred- ators are more complicated. Intuitively, we might " #kt and b cVcW . Conditions for the existence expect that increasing the level of habitat destruc- of E*G are provided in Appendix A. The Jacobian tion should always lead to a smaller fraction of
JTBI 20012304 PREDATOR}PREY METAPOPULATION 7
FIG. 2. A depiction comparable to Fig. 1, except in this k' case cV. Note that the equilibrium density of prey in- creases as habitat destruction increases, until the point of predator extinction. The patterns with respect to resource supplementation are consistent with those from Fig. 1.
FIG. 1. The fraction of sites occupied at equilibrium for prey (solid line) and predator (dashed line) as predicted by due to co-occurrence of predators and prey on the ignorant-predator metapopulation model (3). In this k( a patch is less than the probability of coloniz- case, cV. Note that prey decline monotonically as habi- tat destruction increases, but at a faster rate after extinction ation by prey of a vacant, habitable patch; i.e., k( of predators. Also, note the dramatic positive impact of cV (Appendix A). In this case the equilibrium resource supplementation on predators, and concomitantly fraction of prey patches, x*, declines linearly its negative impact on prey of type X. Resource supple- mentation is indexed by t, with lower values indicating with increasing habitat destruction (Fig. 1). k' greater levels of resource supplementation, or equivalently, When cV the situation is reversed and the less reliance on X-type prey for survival. per-patch &&death'' rate due to predation exceeds the per-patch &&birth'' rate due to colonization (Appendix A). In this case prey actually bene"t patches occupied by prey. However, this is not from habitat destruction, because the reduction true, because under certain circumstances the ef- in the fraction of patches occupied by predators fect of habitat destruction is less detrimental than increases predator-free patches faster than the e!ect of predation. Speci"cally, the cost to patches are destroyed. Thus, prey &&escape'' from prey of predation is less than the cost of habitat predation is facilitated by habitat loss, and x* destruction when the probability of extinction increases linearly with D (Fig. 2). For both cases,
JTBI 20012304 8 R. K. SWIHART E¹ A¸. when the predator su!ers extinction the domain values as switches to E, and the slope of x* changes ac- cordingly (Figs 1 and 2). k(e #t) c " W Resource supplementation by predators also W k! ! # ( cV)(1 D) eV leads to a range of predator}prey equilibrial rela- tionships as a function of habitat destruction. and The fraction of patches occupied by prey at equi- c e #t(c D#e ) librium always exceeds the fraction occupied by c " V W V V . W c (1!D) specialist predators (Figs 1 and 2), consistent with V the models of May (1994) and Bascompte & SoleH Note that c (0 when k(c !(e /1!D), (1998). However, for low to moderate levels of W V V whereas c '0 when k'c !(e /(1!D)). Also habitat destruction generalist predators can oc- W V V note that c '0 over the feasible range of para- cupy a greater fraction of patches at equilibrium W meter values. Consequently, when k'c coexist- than X-type prey (Figs 1 and 2). And when pred- V ence occurs if and only if c (c (c . Outside ators are so generalized in their resource use that W W W of this range, either predators only (c 'c )or they need not rely on X-type prey other than W W prey only (c (c ) exist (Fig. 4). When k(c , incidentally, y*'x* at all levels of habitat de- W W V only prey can occur for c (c , and coexistence struction, provided that k(c (Fig. 1). W W V occurs for c 'c . A fundamental outcome of the ignorant-pred- W W The equilibrium fraction of patches occupied ator model is that resource supplementation by by predators exhibits a nonlinear response to predators reduces equilibrial levels of prey predator colonization rate, and the position and occupancy of patches (Figs 1 and 2). By relying severity of the threshold varies as a function of on bu!er prey, generalist predators are able habitat destruction and resource supplementa- to persist in patches without X-type prey tion (Fig. 3). In general, habitat destruction while simultaneously using these patches as sour- increases the colonization rate necessary for ces of colonists for patches containing X-type predator persistence in a landscape. Specialist prey. predators are much more severely a!ected by habitat loss, both in terms of the threshold level 6.3. EFFECTS OF PREDATOR COLONIZATION RATE of colonization required for persistence and in terms of the equilibrium occupancy attained Bascompte & SoleH (1998) demonstrated thre- (Fig. 3). As the per-patch probability of prey ex- sholds for c , the per-patch colonization rate of W tinction due to predation (i.e. k) increases, the predators. For rates below a threshold value, equilibrium density of predators declines because predators su!ered extinction, whereas small in- fewer patches contain X-type prey. For a given creases in colonization rate above the threshold level of habitat destruction, increases in the prob- resulted in a rapid increase in the equilibrium ability of extinction of X-type prey due to pred- fraction of patches occupied by predators. We ation have a greater negative impact on specialist analysed the ignorant-predator model (3) to de- predators (Fig. 3). termine how the equilibrium patch density of predators, y*, was a!ected by cW, and speci"cally to ascertain whether threshold behavior was 6.4. COMPARISON OF ANALYTICAL AND exhibited. SIMULATION MODELS Unlike the model (1) of Bascompte & SoleH After comparing their analytical model and (1998), ignorant predators [Eqn (3)] can persist in cellular automata, Bascompte & SoleH (1998, a landscape even in the absence of X-type prey p. 391) concluded that the predictions made by (i.e. E). Thus, two critical values are required to the two approaches were similar, although &&mi- determine the range of cW over which coexistence nor di!erences arise as a consequence of real occurs. Let c and c be the predator coloniz- space e!ects''. However, inspection of a subset of W W ation rates at which x*"0 and y*"0, respec- their results suggests that di!erences can be tively. Speci"cally, we can express these critical substantial. We have illustrated their simulation
JTBI 20012304 PREDATOR}PREY METAPOPULATION 9
FIG. 4. A comparison of analytical predictions from the mean "eld model and simulation results from the spatially explicit model of Bascompte & SoleH (1998). Note the large discrepancy in results for the predator equilibria from the two models. The parameter values are taken from Fig. 7 of Bascompte & SoleH (1998): k"0.5, c "0.4, c "0.7, " " V W eV eW 0.2: (22) Prey; (**) Predator.
are more restrictive for a spatially explicit model than for an equivalent mean "eld model. We believe that much of the discrepancy between results of the spatially structured model and the FIG. 3. The fraction of sites occupied at equilibrium by mean "eld model, as well as the apparent contra- the predator as a function of its colonization rate, for the ignorant-predator model. Resource supplementation has diction with the "ndings of Sato et al. (1994), a dramatic impact on the equilibrium density of predators, arises from di!erences between the nominal col- and the critical colonization rate necessary for predators to onization rates, cG, of the analytical models and persist is related in a nonlinear fashion to D and t. The solid dots represent critical rates of colonization, c , above which the e!ective colonization rates, cG, of the spatially W only predators exist. This condition only occurs when explicit models (see below). k'c !e /(1!D). Parameter values are c "0.7, e " Colonization rates are constants in the analyti- " V V k" k" V V eW 0.1: (**) 0.2; (**) 0.8. cal models (1) and (2). They represent the prob- ability of settlement of a vacant, habitable patch, and this probability is independent of the status results and superimposed their analytical model's of neighbouring patches. In contrast, e!ective corresponding predictions for a set of parameter colonization rates in the spatially explicit models values used in their study (Fig. 4). In an intact are determined by both the nominal colonization landscape, equilibrial densities of predator and rate and by the status of neighboring patches, prey are considerably greater than predicted by which in turn is determined by the occupancy of their analytical model; the increase for predators species i. Thus, the e!ective colonization rate is nearly an order of magnitude. In addition, both varies both spatially and temporally. As a "rst species persist in a spatially structured landscape approximation, assume that the probability of at much greater levels of habitat destruction than occupancy of neighboring patches follows a bi- predicted by their analytical model (Fig. 4). Initial nomial distribution. Then for the case of four comparisons of the ignorant-predator model and nearest-neighbor patches, its spatially explicit counterpart also suggested di!erences. In a single-species system, such as exists after extinction of predators, Sato et al. " + 4 L ! \L ! ! L cG A B k (1 k) (1 (1 cG) ), (6) (1994) have shown that conditions for persistence L n
JTBI 20012304 10 R. K. SWIHART E¹ A¸. where k is the fraction of all possible patches occupied by species i and n is the number of neighboring patches occupied by species i. For a "xed k, an increase in the nominal rate of colonization increases the e!ective rate of colon- ization because an occupied neighboring patch is more likely to serve as a source of colonists. Likewise, for a "xed cG, an increase in the overall density of occupied patches increases the e!ective rate of colonization because more neighboring patches are likely to be occupied on average. In our spatially explicit model, c represents the probability of an empty patch being colonized only if it has a single occupied neighbor. In con- trast, c in the mean "eld model is independent of local spatial or temporal variation in patch occu- pancy. The di!erence is important, because it captures a critical biological feature of spatially explicit systems, namely, distance and density e!ects on colonization processes. To compare our analytical and simulation re- sults, we calculated e!ective colonization rates from eqn (6) for each steady state produced by the simulation model. These e!ective colonization rates were then used in eqn (5) to compute equilib- rial values for predator and prey under the ignor- ant-predator model. If coexistence failed to occur, the appropriate boundary equilibria were used. A substantial quantitative improvement was made when comparing simulation results to ana- lytical predictions based on e!ective colonization rates (e.g. Fig. 5) as opposed to nominal coloniz- ation rates (Fig. 1). Simulation results for predators agreed reasonably well with analytical FIG. 5. A comparison of results from the ignorant-pred- ator model (3) with its spatially explicit counterpart (circles), predictions, although the predictions consistently k( for the case where cV. The e!ective colonization rate, cG, were better for generalist predators than for was used to compute predicted equilibrial values for the specialists. Predators responded to changes in analytical model, as described in the text. Thus, the equilib- resource supplementation as predicted, whereas rial values for the ignorant-predator model are greater than k( those in Fig. 1, where the nominal colonization rates, cG, prey did not (Figs 5 and 6). When cV, predic- were used. Parameter values are the same as those used in tions for prey were qualitatively comparable to Fig. 1. simulation results (Fig. 5). However, when k' cV, predictions and simulation results for prey matched poorly (Fig. 6). In all instances, restricted dispersal leads to an occupancy pattern spatial structure prolonged the coexistence of spe- for neighboring patches that is more aggregated cies when confronted with habitat destruction. than a binomial distribution. Rather, restricting The disparities between results of the analyti- colonization to neighboring patches leads to aggre- cal and simulation models are attributable, at gations of patches containing predators and prey least in part, to the inclusion of spatial structure (Bolker & Pacala, 1997). Our simulations begin and of discrete time steps in the latter (Durrett with random spatial patterns, but local aggrega- & Levin, 1994). The spatial structure imposed by tions o!er high probabilities of recolonization
JTBI 20012304 PREDATOR}PREY METAPOPULATION 11
though they become extinct in the succeeding iteration. This e!ect of discrete time steps, and the resulting departure from analytical predic- tions, becomes more pronounced as the probabil- ity of extinction increases. That is, as the expected duration of persistence decreases in the analytical model, the impact of persisting for one additional time period in the discrete version is more pro- nounced. Thus, the di!erences between our dis- crete and continuous time models are greater for specialist than for generalist predators. After asu$cient period of time has elapsed, the frac- tion of sites occupied by predator and prey at- tains a steady state. However, the spatial pattern of predator and prey continues to shift across the landscape. Such shifts are emergent properties of spatially structured models of interacting popula- tions with restricted dispersal (Keitt & Johnson, 1995; Bolker & Pacala, 1999). One important and non-intuitive consequence of spatial structure and discrete time was the promotion of coexistence of predator and prey over a wider range of habitat destruction than predicted by our analytical results (Figs 5 and 6). Similarly, spatial heterogeneity has been shown to increase coexistence of species in theoretical
FIG. 6. A comparison of results from the ignorant-pred- (Keitt, 1997) and experimental (Hu!aker, 1958) ator model (3) with its spatially explicit counterpart (circles), food webs. k' for the case where cV. The e!ective colonization rate, cG, was used to compute predicted equilibrial values for the analytical model, as described in the text. Thus, the equilib- 7. Summary and Discussion rial values for the ignorant-predator model are greater than those in Fig. 2, where the nominal colonization rates, cG, The ignorant-predator model (2) extends the were used. Parameter values are the same as those used in study of predator}prey metapopulations by in- Fig. 2. corporating resource supplementation. Equilib- rial densities for coexisting species are always following extinctions, and these local aggrega- stable for the ignorant-predator model, whereas tions can be quite persistent. Declines in occu- instability commonly occurs for the model of pancy rate with increased destruction in the spa- Bascompte & SoleH (1998). The two models pro- tially explicit model are more gradual and linear duced comparable results in some ways, but not than those in the analytical model, presumably in others. We highlight these comparisons below due to the non-random clustering of predator by expanding on some of the conclusions reached and prey in the former (Figs 5 and 6). by Bascompte & SoleH (1998): Discrete time steps in the spatially explicit (1) Specialist predators are driven extinct by model permit prey to escape extinction even lower values of habitat destruction than prey. when predators are common and widespread by However, resource supplementation counteracts incorporating a time lag into the dynamics. Prey this e!ect, and generalist predators can be less can safely colonize sites containing predators, sensitive to habitat loss than the focal prey with no ill e!ects incurred until the following species. time step. Similarly, specialist predators are al- (2) The equilibrium fraction of sites occupied lowed to invade patches without prey, even by the predator exhibits a nonlinear response to
JTBI 20012304 12 R. K. SWIHART E¹ A¸. reductions in their colonization rate. This thre- habitat loss and fragmentation (Hagan & Joh- shold response is more pronounced for generalist nston, 1992; Johnson & Schwartz, 1993), and than for specialist predators. Conversely, general- recent evidence suggests that generalist predators ist predators are more capable of persisting when may contribute signi"cantly to the problem their colonization rates are low. (Keyser et al., 1998; Gehring & Swihart, unpubl. (3) Following extinction of predators, the data). Increased destruction of arti"cial nests of negative e!ect of additional habitat loss on re- tetraonids due to generalist avian predators also gional prey abundance is intensi"ed. has been linked to habitat fragmentation in Fen- (4) Although the equilibrium fraction of sites noscandia (AndreH n et al., 1985). Thus, our results occupied by prey is reduced due to predation, the suggest that increased attention should be e!ects of predation and habitat destruction on focussed on the fate of prey species subjected to prey are complementary. When the risk of local predation by generalist species which have extinction due to predation exceeds the rate at adapted well to the loss or degradation of native which patches are colonized, habitat destruction habitat. can actually increase the equlibrial fraction of Our results also predict that prey colonization sites occupied by prey. rate and the risk of prey extinction due to pred- (5) Our reanalysis suggests that substantial ation interact in a non-intuitive manner to a!ect di!erences can occur between the predictions of the equilibrial densities of prey. High risk of ex- the analytical model of Bascompte & SoleH (1998) tinction due to predation (relative to prey colon- and their spatially explicit stochastic model. ization rate) depresses the equilibrium fraction of Much of the di!erences can be attributed to patches occupied by both species. However, the a constant, nominal colonization rate in the ana- e!ect of habitat destruction on equilibrial density k' lytical model versus a distance- and density-de- is less severe for both species when cV, and pendent colonization rate in the spatially explicit prey can even bene"t under these circumstances formulation. Additional di!erences are due to (Fig. 2). The risk of prey extinction is in#uenced endogenous patterns of patch occupancy and by the functional and numerical response of the time lags in spatially explicit models. predator at a local level. Predator responses in Modeling e!orts to date have focused on the turn are linked to mobility (de Roos et al., 1998), e!ects of habitat destruction on specialist pred- and presumably to determinants of niche breadth ators (May, 1994; Kareiva & Wennergren, 1995; and population growth (Wol!, 1999). From the Nee et al., 1997; Bascompte & SoleH , 1998). perspective of prey, colonization rate is in- Certainly, these e!orts have been justi"ed, as #uenced most notably by niche breadth, or the the negative impacts of habitat loss on top pred- ability to use resources in the altered habitat ators are well established (see Belovsky, 1987; surrounding patches (Hansson, 1991; AndreH n, Hoogesteijn et al., 1993; Hunter, 1996). In many 1994; Wol!, 1999). Thus, future studies should landscapes, though, human degradation and explore the relation between the risk of prey alteration of native habitat have occurred for extinction due to predation and the niche centuries. In addition, top predators may be breadth of prey and predator. persecuted and subjected to extirpation before The level of spatial detail to include in habitat destruction becomes important (e.g., a modeling endeavor is an important considera- Palomares et al., 1995). Under either of these tion that can a!ect conclusions about the system scenarios, generalist predators are likely to prolif- being studied (Durrett & Levin, 1994). In our erate at the expense of specialists. Our results analytical formulation, a principal objective was suggest that in landscapes already subjected to to extend the model proposed by Bascompte disturbance, prey species may be more imperiled & SoleH (1998) to allow for resource supplementa- than predators. This is particularly true for prey tion. Thus, we used a pair of ordinary di!erential which serve solely as an incidental source of sus- equations, or mean "eld approach, for consist- tenance for predators. For instance, populations ency with their earlier work. We also introduced of ground-nesting songbirds in grassland habita- spatial structure explicitly into the system by ts of the central United States have su!ered from means of our cellular automaton. Although our
JTBI 20012304 PREDATOR}PREY METAPOPULATION 13 main conclusions were una!ected by the level of constructing the ignorant-predator model of pred- model detail chosen, interesting di!erences arose ator}prey dynamics. We appreciate the e!orts of in some characteristics of the system. For in- George S. McCabe, who was instrumental in promo- ting collaboration among the authors. This is paper stance, the spatially explicit approach revealed number 99-5322 of the Purdue University Agricul- the role of endogenous patterns of patch occu- tural Research Programs. pancy that cannot be shown in the mean "eld model. For species characterized by long-dis- REFERENCES tance dispersal, such as some pelagic-spawning ANDERSEN, O., CROW, T. R., LIETZ,S.M.&STEARNS,F. "shes (Moyle & Cech, 1996), the mean "eld (1996). Transformation of a landscape in the upper mid- model may be a more appropriate framework west, U.S.A.: the history of the lower St. Croix river than a spatially structured model. However, valley, 1830 to present. ¸andscape ;rban Planning 35, 247}267. attention to the di!erences between the two ANDRED N, H. (1994). E!ects of habitat fragmentation on approaches certainly is warranted in biological birds and mammals in landscapes with di!erent propor- systems characterized by restricted dispersal rela- tions of suitable habitat: a review. Oikos 71, 355}366. tive to the scale at which metapopulation persist- ANDRED N, H., ANGELSTAM, P., LINDSTROG M,E.&WIDED N,P. (1985). Di!erences in predation pressure in relation to ence is measured. Although beyond the scope of habitat fragmentation: an experiment. Oikos 45, 273}277. this paper, we believe that such attention in the BASCOMPTE,J.&SOLED , R. V. (1998). E!ects of habitat future could be applied toward developing fully destruction in a prey}predator metapopulation model. J. theor. Biol. 195, 383}393. spatial stochastic analytical models. Recently, in- BELOVSKY, G. E. (1987). Extinction models and mammalian terspeci"c competition models of this type have persistence. In:
JTBI 20012304 14 R. K. SWIHART E¹ A¸.
GAINES, M. S., DIFFENDORFER, J. E., TAMARIN,R.H. MAY, R. M. (1994). The e!ect of spatial scale on ecological &WHITTAM, T. S. (1997). The e!ects of habitat fragmenta- questions and answers. In: ¸arge-Scale Ecology and tion on the genetic structure of small mammal popula- Conservation Biology (Edwards, P. J., May, R. M. & tions. J. Heredity 88, 294}304. Webb, N. R., eds), pp. 1}17. Oxford: Blackwell Scienti"c GURNEY,W.S.C.&NISBET, R. M. (1998). Ecological Publications. Dynamics, 335pp. New York: Oxford University Press. MOILANEN,A.&HANSKI, I. (1995). Habitat destruction and HAGAN,J.M.&JOHNSTON, D. W. (1992). Ecology and coexistence of competitors in a spatially realistic meta- Conservation of Neotropical Migrant ¸andbirds. Washing- population model. J. Anim. Ecol. 64, 141}144. ton, DC: Smithsonian Institution Press. MOG NKKOG NEN,M.&REUNANEN, P. (1999). On critical thre- HANSKI, I. (1998). Metapopulation dynamics. Nature 396, sholds in landscape connectivity: a management perspect- 41}49. ive. Oikos 84, 302}305. HANSKI,I.&KORPIMAKI, E. (1995). Microtine rodent dy- MOYLE, P.B. & CECH, Jr., J. J. (1996). Fishes: An Introduc- namics in northern Europe: parameterized models for the tion to Ichthyology, 3rd Edn. Upper Saddle River, New predator}prey interaction. Ecology 76, 840}850. Jersey: Prentice Hall. HANSSON, L. (1991). Dispersal and connectivity in meta- NEE, S., MAY,R.M.&HASSELL, M. P. (1997). Two species populations. Biol. J. ¸innean Soc. 42, 89}103. metapopulation models. In: Metapopulation Biology. Ecol- HECNAR,S.J.&M'CLOSKEY, R. T. (1997). Patterns of ogy, Genetics and Evolution (Hanski, I. & Gilpin, M., eds), nestedness and species association in a pond-dwelling am- pp. 123}147. San Diego: Academic Press. phibian fauna. Oikos 80, 371}381. NUPP,T.E.&SWIHART, R. K. (1996). E!ect of forest patch HOLYOAK,M.&LAWLER, S. P. (1996). Persistence of an area on population attributes of white-footed mice extinction-prone predator}prey interaction through meta- (Peromyscus leucopus) in fragmented landscapes. Canadian population dynamics. Ecology 77, 1867}1879. J. Zool. 74, 467}472. HOOGESTEIJN, R., HOOGESTEIJN,A.&MONDOLFI,E. NUPP,T.E.&SWIHART, R. K. (1998). E!ects of forest (1993). Jaguar predation and conservation: cattle mortality fragmentation on population attributes of white- causes by felines on three ranches in the Venezuelan footed mice and eastern chipmunks. J. Mammal. 79, Llanos. In: Mammals as Predators (Dunstone, N. & Gor- 1234}1243. man, M. L., eds), pp. 391}407. Oxford: Clarendon Press. PALOMARES, F., GAONA, P., FERRERAS,P.&DELIBES,M. HUFFAKER, C. B. (1958). Experimental studies on predation: (1995). Positive e!ects on game species of top predators by dispersion factors and predator}prey oscillations. Hilgar- controlling smaller predator populations: an example with dia 27, 343}383. lynx, mongooses, and rabbits. Cons. Biol. 9, 295}305. HUNTER Jr, M. L. (1996). Fundamentals of Conservation SATO, K., MATSUDA, H., & SASAKI, A. (1994). Pathogen Biology. Cambridge: Blackwell Science. invasion and host extinction in lattice structured popula- HUXEL,G.R.&HASTINGS, A. (1998). Population size de- tions. J. Math. Biol. 32, 251}268. pendence, competitive coexistence and habitat destruction. SAUNDERS, D. A., HOBBS,R.J.&MARGULES, C. R. (1991). J. Anim. Ecol. 67, 446}453. Biological consequences of ecosystem fragmentation: a re- JOHNSON,D.H.&SCHWARTZ,M.D.(1993).Theconservation view. Cons. Biol. 5, 18}32. reserve program and grassland birds. Cons. Biol. 7, 934}937. SCHMIDT,K.A.&WHELAN, C. J. (1998). Predator-mediated KAREIVA,P.&WENNERGREN, V. (1995). Connecting interactions between and within guilds of nesting song- landscape patterns to ecosystem and population process. birds: experimental and observational evidence. Am. Nat. Nature 373, 299}302. 152, 323}402. KEITT, T. H. (1997). Stability and complexity on a lattice: SHEPERD,B.F.&SWIHART, R. K. (1995). Spatial dynamics coexistence of species in an individual-based food web of fox squirrels (Sciurus niger) in fragmented landscapes. model. Ecol. Modelling 102, 243}258. Canadian J. Zool. 73, 2098}2105. KEITT,T.H.&JOHNSON, A. R. (1995). Spatial heterogeneity TILMAN, D., MAY,R.M.LEHMAN,C.L.&NOWAK,M.A. and anomalous kinetics: emergent patterns in di!usion- (1994). Habitat destruction and the extinction debt. Nature limited predatory}prey interaction. J. theor. Biol. 172, 371, 65}66. 127}139. VICKERY, P. D., HUNTER,M.L.&WELLS, J. F. (1992). KEYSER, A. J., HILL,G.E.&SOEHREN, E. C. (1998). E!ects Evidence of incidental nest predation and its e!ect on nests of forest fragment size, nest density, and promixity to edge of threatened grassland birds. Oikos 63, 281}288. on the risk of predation to ground-nesting passerine birds. WILCOX,B.A.&MURPHY, D. D. (1985). Conservation Cons. Biol. 12, 986}994. strategy: the e!ects of fragmentation on extinction. Am. KOLOZSVARY,M.B.&SWIHART, R. K. (2000). Habitat Nat. 125, 879}887. fragmentation and the distribution of amphibians: patch WITH, K. A., GARDNER,R.H.&TURNER, M. G. (1997). and landscape correlates in farmland. Canadian J. Zool., in Landscape connectivity and population distributions in press. heterogeneous environments. Oikos 78, 151}169. LEVINS, R. (1969). Some demographic and genetic conse- WITH,K.A.&KING, A. W. (1999). Extinction thresholds for quences of environmental heterogeneity for biological con- species in fractal landscapes. Cons. Biol. 13, 314}326. trol. Bull. Entom. Soc. Am. 15, 237}240. WOLFF, J. O. (1999). Behavioral model systems. LIMA,S.L.&ZOLLNER, P. A. (1996). Towards a behavioral In: ¸andscape Ecology of Small Mammals (Barrett, G. W. ecology of ecological landscapes. ¹rends Ecol. Evolution &PELES, J. D., eds), pp. 11}40. New York: 11, 131}135. Springer-Verlag. MASON,D.M.&PATRICK, E. V. (1993). A model for the ZOLLNER, P. A. (2000). Comparing the landscape-level per- space-time dependence of feeding for pelagic "sh popula- ceptual abilities of forest sciurids in fragmented agricul- tions. ¹rans. Am. Fish. Soc. 122, 884}901. tural landscapes. ¸andscape Ecol., 15, 523}533.
JTBI 20012304 PREDATOR}PREY METAPOPULATION 15
APPENDIX A and because v"(1!x)y, i.e., the fraction of Equivalent Formulation for Ignorant patches occupied by predator only, Predator Model dy dv dz Here we demonstrate the equivalence of our " # "c y(1!y!D)!e y!t(1!x)y. formulation for the ignorant predator model with dt dt dt W W a formulation focusing on state-transitions of patches. In addition to the terminology already These are eqns (3a) and (3b). introduced, let u"prey-only patches, v"pred- ator-only patches, and z"patches with both Stability Analysis species. Then x"u#z, y"v#z, v"(1!x)y and z"xy. The system of ordinary di!erential We "rst examine the stability of E* for the equations for u, v, and z can be written as follows: model of Bascompte & SoleH (1998). The Jacobian of E* is given by du" # ! ! ! ! cV(u z)[1 u v z D] c (1!D)!e !2c x* !k dt J" V V V . A c y* !c y*B ! ! # # W W eVu cWu(v z) eWz. " (C# k Note that Det(J) cWy* 4cV eW/cW, which The bracketed term refers to empty patches. Also, is always positive (consult the text for a de"nition colonization of prey-only patches by predators of C ). Hence, E* is stable if the Tr(J)(0, and changes the patch from state u to z (third term) unstable if Tr(J)'0 (Gurney & Nisbet, 1998). and extinction of predators from z-type patches De"ne a critical value of D, DA, such that changes them to state u (fourth term). The equa- D "D !(e /c ). Then Tr(J)(0 can be rewrit- A A W W tion for dv/dt is ten as
dv ! k( "c (v#z)[1!u!v!z!D] (DA D) dt W (c (D !D)#e )(c (D !D)#c (D !D)) V A W V A W A !e v!tv!c v(u#z)#e z#kz. # . W V V 2cV cW The last term describes the transition of a patch (A.1) with both species (z) to a patch with predators Recall that E* exists only if D(D , i"1, 2. only (v) due to predation. Finally the equation for A dz/dt is Thus, the quantity on the right-hand side of the inequality (A.1) is positive. Now de"ne a critical k k value of , A, such that dz "c u(v#z)#c v(u#z)!(e #e #k)z. k "f (D)" dt W V V W A (c (D !D)#e )(c (D !D)#c (D !D)) The last term describes transitions out of state V A W V A W A . (D !D)(2c #c ) z due to &&intrinsic'' death rates and to predation. A V W It follows from the identities above that, because ' ( z"xy, i.e., the fraction of patches occupied by We can show that Tr(J) 0 for D DA and k'k both predator and prey, A. For the parameter values used in Fig. 3 of Bascompte & SoleH (1998), D "0.514 (thus de"n- A ing an upper limit for the existence of E*), " k DA 0.403, and the form of A is illustrated in dx"du#dz" ! ! ! !k Fig. A1 along with the regions of existence and cVx(1 x D) eVx xy k dt dt dt instability of E* in (D, A) space.
JTBI 20012304 16 R. K. SWIHART E¹ A¸.
E* for the ignorant predator model exists if and only if D(D and D(D . Conditions for V W the stability of E*G are given in the text.
E4ects of Resource Supplementation and Habitat Destruction Next, we turn our attention to the e!ects of t and D on x* and y* in the ignorant-predator model. Consider x* and y* as functions of t and D, denoted F(t, D) and G(t, D), respectively. Note that
FIG. A1. Critical values of k and D in relation to regions * F t " 1 of instability for the interior equilibria of the model for- *t ( , D) #kt mulated by Bascompte & SoleH (1998). The portion of the (cVcW ) curve to the right of D is plotted only for completeness, as it A lies beyond the region of coexistence. Parameter values are k"0.6, c "0.4, c "0.9, e "0.15, e "0.1. ; k # #k ! !eW ' V W V W A cVcWD cWeV cWA1 D BB 0 cW
For the ignorant-predator model, we examine and the stability properties of E by noting that the Jacobian at E is *G !1 (t, D)" *t (c c #kt) !c x !kx V W J " V . A ! ! !t # B 0 cW(1 D) eW (D (eV/cV)) e ; c D#e )c c #kc c 1!D! W (0, A V V V W V WA c BB Because the eigenvalues are represented by the W diagonal elements of J, E is stable only if ! ! !t # ( for 0)D(D and 0)D)1. Thus, for any cW(1 D) eW (D (eV/cV)) 0. In terms of V D, E is unstable if D(D , where D is given by "xed value of habitat destruction, x* increases V V with t, albeit at a declining rate, whereas y* is t c c !c e !e t negatively related to , with the rate of change D " V W V W V . becoming less negative as t increases, In a similar V c (c #t) V W fashion, we can examine the in#uence of D on x* and y* by noting that The Jacobian at E for the ignorant-predator model is *F c (k!c ) (t,D)" W V " *D c c #kt J V W c (1!D)!e !k(1!D!e #t/c ) 0 and V V W W . A t ! B y cWy *G c (c #t) (t,D)" V W . In analogous fashion to E, E is stable only if *D c c #kt ! ! !k ! ! #t ( V W cV(1 D) eV (1 D (eW )/cW) 0. In terms of D, E is unstable if D(D , where W Thus, for any "xed value of t, x* increases with k' k( D if cV and decreases with D if cV. c e !k(e #t) y* D "1! W V W . In contrast, always decreases with increas- W !k cW(cV ) ing D.
JTBI 20012304 PREDATOR}PREY METAPOPULATION 17
) ) State Transitions for the Spatially Explicit Model prey and predator, respectively (0 nG 4). Let the four states of habitable patches be Then the following matrix of i rows and j represented by 0 (empty), 1 (prey only), 2 (pred- columns represents the entire set of tran- ator only), and 3 (both species). Further, let sition probabilities, assuming that extinction p represent the probability of transition from and recolonization events for a single patch GH do not both occur within a given time state j to state i. Finally, let nV and nW represent the number of neighboring patches occupied by step:
! LV ! LW ! LW #t ! LV #k #t (1 cV) (1 cW) eV(1 cW) (eW )(1 cV) (eV )(eW ) (1!(1!c )LV)(1!c )LW (1!e )(1!c )LW (1!(1!c )LV)(e #t)(1!e !k)e V W V W V W V W . ! LV ! ! LW ! ! LW ! LV ! !t #k ! !t (1 cV) (1 (1 cW) ) eV(1 (1 cW) )(1cV) (1 eW )(eV )(1 eW ) ! ! LV ! ! LW ! ! ! LW ! ! LV ! !t ! !k ! (1 (1 cV) )(1 (1 cW) )(1 eV)(1 (1 cW) )(1 (1 cV) )(1 eW )(1 eV V)(1 eW)
JTBI 20012304 AUTHOR QUERY FORM
HARCOURT PUBLISHERS
JOURNAL TITLE: JTBI DATE:30/03/2001 ARTICLE NO. : 20012304
Queries and / or remarks Manuscript Details required Author's response Page/line
27 Kolozsvary & Swihart (2000) Please update